326 XV NORMED ALGEBRAS AND SPECTRAL THEORY (use Holder's inequality). Deduce that the function /?i— >Dp(/x) is continuous on the right on ]0, + oo [. (Use the fact that this function is upper semicontinuous.) If /x(X) = 1, then 0 < Dp(ft) <; 1 for 0 < p < + oo. If p> 1 and if fe &p(fj,) and #e JS?e(/x) (where l/p+ l/q= 1) are positive functions, then • ft) ^ (D,(/p • ft))1" • (D,(0« • ft))1". If ft(X) = 1, then for each positive function /e -^(/x), we have (b) If Dp(/x) = 1 for some p> 0, show that ft is a representative measure for ^, (Observe that, if veB is such that xW = °» then x(exp(to/p)) = 1 for a^ f>0; show that \&tv dp ^ 0 by making / tend to 0, and deduce that I &v dp, = 0.) (c) Conversely, if /x is a representative measure for #, then Dp(/x) =1 for all p ^ 1. Show that Dp(ft) = 1 for all p > 0 if and only if ft is a Jensen measure for % (use Section 13.11, Problem 12(e)). (d) Let/? be such that 1 ^p < 4- oo and Dp(/x) > 0. Show that there exists a function h e &*(& (where (1/p) 4- (I/?) = 1) such that N9(/i) = 1 and for all w e B. (Use the Hahn-Banach theorem, Proposition (13.17.1) and Problem 1 of (13.17).) If p > 1, the measure \h\q - p is a representative measure for ^. 11. With the hypotheses and notation of Problem 8, for every function/^ 0 belonging to ^(ft), put !„(/) = exp[ I log |/| dfj\ (with exp(- oo) = 0). A representative measure /x for x is said to be hyperextremal if every representative measure for x with base //. is necessarily equal to /x ; it is then an extremal point of the convex set $(x) (Problem 9). (a) Let;? ^ 1. For a positive measure p of mass 1 on X to be a Jensen measure for X, it is necessary and sufficient that Dp(/- p,) J> J^(/) for all functions/ ^ 0 belonging to ^((j) .(To show that the condition is sufficient, consider the function (\u\ + e)~p, where u e B and e > 0.) (b) Let p ^ 1 and let /x be a positive measure of mass 1 on X. Then the following conditions are equivalent : (a) /x is a hyperextremal representative measure for x- (ft) ^ is a representative measure for ^, and Dp(/- ft) g J^(/) for all functions /;> 0 belonging to ^^/x). (y) /x is a representative measure for x> anc^ f°r each/e ^'(ft) such that/- ft is a quasi-representative measure for #» we have l/Vft £ Wl/l). Moreover, if these conditions are satisfied, then ft is a Jensen measure for #, and D//- ft) = JM(/)for all^ ^ 1 and all functions/^ 0 belonging to JS?1^)- (To show that (a) implies ($), prove that if ft is hyperextremal and p ^ 1 and /;>0 belongs to -^"(ft), then (Dp(/p • ft))1/p g Dj(/- ft): for this, use Problem 10(d). Then use Problem 10(a) to deduce that iff,g are ;>0 and belong to -^(ft), then &i(fff ' ft) ^ Di(/- ft)D!^ • /x). Conclude that, for all functions /g> 0 in ^(ft), we).n particular, the image of C under the